By Witold Pedrycz
- A entire insurance of rising and present know-how facing heterogeneous resources of data, together with info, layout tricks, reinforcement indications from exterior datasets, and comparable topics
- Covers all helpful necessities, and if necessary,additional reasons of extra complicated themes, to make summary techniques extra tangible
- Includes illustrative fabric andwell-known experimentsto supply hands-on experience
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Additional info for Knowledge-Based Clustering: From Data to Information Granules
In contrast to the possibility measure, the necessity measure is asymmetric (which is obvious, as we are concerned with the inclusion predicate). 2. Computations of possibility (a) and necessity (b) measures; t-norm: minimum, s-norm: maximum. The dotted line in (b) shows a complement of A, 1 − A(x). 36 FUZZY SETS AND FUZZY RELATIONS By its nature (as a measure of overlap), the possibility measure is symmetric, Poss(A, X) = Poss(X, A). The necessity measure, expressing the extent of inclusion, is not symmetric.
O. E. G. Stork, Pattern Classiﬁcation, 2nd edition, John Wiley, New York, 2001. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. of Cybernetics, 3, 3, 1974, 32–57. H. Frigui, R. Krishnapuram, A comparison of fuzzy shell clustering methods for the detection of ellipses, IEEE Trans. on Fuzzy Systems, 4, 1996, 193–199. K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd edition, Academic Press, London, 1990. F. Hoppner, F. Klawonn, R.
37) i=1 The weight coefﬁcient δ2 reﬂects the distance between all data and the noise cluster. Note that we end up with c + 1 clusters, with the extra cluster serving as the noise cluster. The difference in the second term of the objective function expresses the degree of membership of each pattern in the noise cluster. The sum over the ﬁrst c is less than or equal to 1. 8. SELF-ORGANIZING MAPS AND FUZZY OBJECTIVE FUNCTION-BASED CLUSTERING Objective function-based clustering forms one of the main optimization paradigms of data discovery.